Comparative Analysis of Wave Scattering Numerical Modeling Using the Boundary Element Method and Physics-Informed Neural Networks

This study compares the Boundary Element Method (BEM) and Physics-Informed Neural Networks (PINNs) for solving the two-dimensional Helmholtz equation in wave scattering problems. The objective is to evaluate the performance of both methods under the same conditions. We solve the Helmholtz equation using BEM and PINNs for the same scattering problem. PINNs are trained by minimizing the residual of the governing equations and boundary conditions with their configuration determined through hyperparameter optimization, while BEM is applied using boundary discretization. Both methods are evaluated in terms of solution accuracy and computation time. We conducted numerical experiments by varying the number of boundary integration points for the BEM and the number of hidden layers and neurons per layer for the PINNs. We performed a hyperparameter tuning to identify an adequate PINN configuration for this problem as a network with 3 hidden layers and 25 neurons per layer, using a learning rate of $10^{-2}$ and a sine activation function. At comparable levels of accuracy, the assembly and solution of the BEM system required a computational time on the order of $10^{-2}$~s, whereas the training time of the PINN was on the order of $10^{2}$~s, corresponding to a difference of approximately four orders of magnitude. However, once trained, the PINN achieved evaluation times on the order of $10^{-2}$~s, which is about two orders of magnitude faster than the evaluation of the BEM solution at interior points. This work establishes a procedure for comparing BEM and PINNs. It also presents a direct comparison between the two methods for the scattering problem. The analysis provides quantitative data on their performance, supporting their use in future research on wave propagation problems and outlining challenges and directions for further investigation.

💡 Research Summary

This paper presents a systematic benchmark comparing the Boundary Element Method (BEM) and Physics‑Informed Neural Networks (PINNs) for solving two‑dimensional Helmholtz‑governed wave scattering problems. The authors choose the canonical case of acoustic scattering by a circular obstacle illuminated by a plane wave. The governing equations consist of the Helmholtz equation in the exterior domain, a Neumann condition on the obstacle surface, and the Sommerfeld radiation condition at infinity.

For the BEM implementation, the authors derive the boundary integral equation using the 2‑D Helmholtz Green’s function (v = \frac{i}{4}H^{(1)}0(kr)). The circular boundary is discretized into a set of elements; Gaussian quadrature is employed on each element to evaluate the influence matrices (G{ij}) and (\hat H_{ij}). Singular integrals (i=j) are treated analytically to preserve accuracy. The resulting linear system is assembled and solved directly, after which the scattered field is evaluated at a uniform grid of interior points.

The PINN approach builds a fully connected multilayer perceptron that maps spatial coordinates ((x,y)) to the real and imaginary parts of the scattered field. The loss function comprises three contributions: (i) the Helmholtz residual in the interior, (ii) the Neumann residual on the obstacle, and (iii) an absorbing‑boundary‑condition (ABC) residual on an artificial truncation boundary. Collocation points for each term are generated via Latin‑hypercube sampling. Hyper‑parameter optimization yields a network with three hidden layers, 25 neurons per layer, a sine activation function, and a learning rate of (10^{-2}). Training is performed with the Adam optimizer on an RTX 4060 GPU and typically requires on the order of (10^{2}) seconds to reach a prescribed tolerance. Once trained, the network can evaluate the field at any interior point essentially instantaneously.

The authors evaluate both methods on two performance axes: accuracy (measured by the (L_2) error against an analytical Mie‑series reference) and computational cost. By adjusting the number of boundary integration points for BEM and the network architecture for PINNs, they ensure that both methods achieve comparable error levels (≈ 10⁻⁴). Under these conditions, BEM assembly and solution take roughly (10^{-2}) seconds, whereas PINN training consumes about (10^{2}) seconds. However, after training, PINN inference for a batch of 10⁴ interior points requires only ≈ 10⁻² seconds, which is two orders of magnitude faster than the post‑processing step of BEM (≈ 1 second for the same batch).

A notable difference lies in the treatment of the radiation condition. BEM inherently satisfies the Sommerfeld condition through the Green’s function, guaranteeing correct far‑field behavior. PINNs rely on an artificial ABC on a truncated outer boundary; consequently, the far‑field decay can be slightly inaccurate, especially when the training points do not densely cover the outer region. The authors suggest that more sophisticated absorbing layers (e.g., PML) or inclusion of far‑field data could mitigate this issue.

The paper concludes that BEM remains the method of choice when high fidelity, exact enforcement of radiation conditions, and immediate solution are required, particularly for problems with simple geometries. PINNs, despite their heavy upfront training cost, offer remarkable flexibility: once trained, they can be queried repeatedly for different frequencies, boundary shapes, or source configurations without re‑solving a linear system. This makes them attractive for parametric studies, real‑time control, or inverse problems.

Future research directions proposed include (1) hybrid schemes where BEM solutions are used to pre‑train or augment PINNs, (2) scaling studies for high‑frequency or highly irregular scatterers, (3) systematic incorporation of advanced absorbing boundary techniques into the PINN loss, and (4) extension to multi‑physics scenarios (e.g., coupled acoustic‑elastic problems). The authors provide all source code in a public GitHub repository to facilitate reproducibility and further exploration.